Viable $f(T)$ models are practically indistinguishable from $\Lambda \mathrm{CDM}$

We investigate the cosmological predictions of several $f(T)$ models, with up to two parameters, at both the background and the perturbation levels. Using current cosmological observations (geometric supernovae type Ia, cosmic microwave background and baryonic acoustic oscillation and dynamical growth data) we impose constraints on the distortion parameter, which quantifies the deviation of these models from the concordance $\Lambda$ cosmology at the background level. In addition we constrain the growth index $\gamma$ predicted in the context of these models using the latest perturbation growth data in the context of three parametrizations for $\gamma$. The evolution of the best fit effective Newton constant, which incorporates the $f(T)$-gravity effects, is also obtained along with the corresponding $1\sigma$ error regions. We show that all the viable parameter sectors of the $f(T)$ gravity models considered practically reduce these models to $\Lambda \mathrm{CDM}$. Thus, the degrees of freedom that open up to $\Lambda \mathrm{CDM}$ in the context of $f(T)$ gravity models are not utilized by the cosmological data leading to an overall disfavor of these models.

Figure 1

Left: The percent difference $(z,b)$ between the numerical solution of Eqs. (16, 35) and the analytical approximations of Eqs. (64, 65) as a function of $z$, for various values of the parameter $b$ for both methods $M_1$ (at first and second order) and $M_2$. Right: The average percent difference $⟨\mathrm{\text{difference}}(b)⟩$ between the numerical solution of Eqs. (16, 35) and the analytical approximations of Eqs. (64, 65) as a function of the parameter $b$. In this case, the average over the redshift is taken in the range $z\in [0,100]$.

Figure 2

Likelihood contours for $\delta {\chi }^2\equiv {\chi }^2-{\chi }_{\min }^2$ equal to 2.30, 6.18 and 11.83, corresponding to $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels, in the $({\Omega }_m,b)$ plane for the ${\Gamma }_0$ growth rate parametrization and the $f_1\mathrm{CDM}$ (left), $f_2\mathrm{CDM}$ (middle) and $f_3\mathrm{CDM}$ (right) models. In all cases the black point corresponds to the best fit. In this plot and in the ones that follow we have set the parameters that are not shown to their best fit values for the corresponding model (see Table ).

Figure 3

Likelihood contours for $\delta {\chi }^2\equiv {\chi }^2-{\chi }_{\min }^2$ equal to 2.30, 6.18 and 11.83, corresponding to $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels, in the $({\Omega }_m,\gamma )$ plane for the ${\Gamma }_0$ growth rate parametrization and the $f_1\mathrm{CDM}$ (left), $f_2\mathrm{CDM}$ (middle) and $f_3\mathrm{CDM}$ (right) models. In all cases the red point corresponds to $({\Omega }_m,\gamma )=(0.272,6/11)$.

Figure 4

Likelihood contours for $\delta {\chi }^2\equiv {\chi }^2-{\chi }_{\min }^2$ equal to 2.30, 6.18 and 11.83, corresponding to $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels, in the $({\gamma }_0,{\gamma }_1)$ plane for the ${\Gamma }_1$ growth rate parametrization and for the $f_1\mathrm{CDM}$ (left), $f_2\mathrm{CDM}$ (middle) and $f_3\mathrm{CDM}$ (right) models. We also include the theoretical $\Lambda \mathrm{CDM}$ $({\gamma }_0,{\gamma }_1)$ values given by ${\Sigma }_1=(6/11,{\gamma }_1(6/11,{\Omega }_{m0,bf}))$ and ${\Sigma }_2=({\gamma }_{0,bf},{\gamma }_1({\gamma }_{0,bf},{\Omega }_{m0,bf}))$.

Figure 5

Likelihood contours for $\delta {\chi }^2\equiv {\chi }^2-{\chi }_{\min }^2$ equal to 2.30, 6.18 and 11.83, corresponding to $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels, in the $({\gamma }_0,{\gamma }_1)$ plane for the ${\Gamma }_2$ growth rate parametrization and for the $f_1\mathrm{CDM}$ (left), $f_2\mathrm{CDM}$ (middle) and $f_3\mathrm{CDM}$ (right) models. We also include the theoretical $\Lambda \mathrm{CDM}$ $({\gamma }_0,{\gamma }_1)$ values given by ${\Sigma }_1=(6/11,{\gamma }_1(6/11,{\Omega }_{m0,bf}))$ and ${\Sigma }_2=({\gamma }_{0,bf},{\gamma }_1({\gamma }_{0,bf},{\Omega }_{m0,bf}))$.

Figure 6

Likelihood contours for $\delta {\chi }^2\equiv {\chi }^2-{\chi }_{\min }^2$ equal to 2.30, 6.18 and 11.83, corresponding to $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels, for the $f_4\mathrm{CDM}$ model in the $({\Omega }_m,{\gamma }_0)$ plane (left) and the $({\gamma }_0,{\gamma }_1)$ plane (middle) and (right). We also include the theoretical $\Lambda \mathrm{CDM}$ $({\gamma }_0,{\gamma }_1)$ values given by ${\Sigma }_1=(11/16,{\gamma }_1(11/16,{\Omega }_{m0,bf}))$ (with the value ${\gamma }_0=11/16$ corresponding to the DGP) and ${\Sigma }_2=({\gamma }_{0,bf},{\gamma }_1({\gamma }_{0,bf},{\Omega }_{m0,bf}))$. As was mentioned in the text, the difference between the DGP (green point) and $f_4\mathrm{CDM}$ (black point) is due to the different $G_{\mathrm{eff}}(z)$, which affects the evolution of the matter density perturbations.

Figure 7

Comparison of the observed and theoretical evolution of the growth rate $f{\sigma }_8(z)=F(z){\sigma }_8(z)$ for the $f_{1–4}\mathrm{CDM}$ models [$f_1\mathrm{CDM}$ (top left), $f_2\mathrm{CDM}$ (top right), $f_3\mathrm{CDM}$ (bottom left), $f_4\mathrm{CDM}$ (bottom right)] and the various growth rate parameterizations. The dotted, dashed and dot-dashed lines correspond to the best fit ${\Gamma }_0$, ${\Gamma }_1$ and ${\Gamma }_2$ parametrizations while the solid black line corresponds to the exact solution of Eq. (19) for $f{\sigma }_8(z)$ for the $\Lambda \mathrm{CDM}$ model for ${\Omega }_m=0.273$ [56].

Figure 8

The evolution of the growth index $\gamma (z)-\frac{6}{11}$ for the $f_{1–4}\mathrm{CDM}$ models [$f_1\mathrm{CDM}$ (top left), $f_2\mathrm{CDM}$ (top right), $f_3\mathrm{CDM}$ (bottom left), $f_4\mathrm{CDM}$ (bottom right)] and the various growth rate parameterizations. The lines correspond to ${\Gamma }_0$ (blue), ${\Gamma }_1$ (green), and ${\Gamma }_2$ (red).

Figure 9

The evolution of the $G_{\mathrm{eff}}(z)$ for the $f_{1–3}\mathrm{CDM}$ models and the various growth rate parameterizations considered in the text, $f_1\mathrm{CDM}$ (top), $f_2\mathrm{CDM}$ (middle), $f_3\mathrm{CDM}$ (bottom), for all three growth rate parametrizations ${\Gamma }_0$ (left), ${\Gamma }_1$ (middle), and ${\Gamma }_2$ (right). The remarkable agreement between $G_{\mathrm{eff}}(z)$ and unity for the $f_2\mathrm{CDM}$ and $f_3\mathrm{CDM}$ models is easily explained by the fact that these models exhibit little deviation from $\Lambda \mathrm{CDM}$, as is easily seen in Table .